Asymptotic properties of correlation bounds in functional spaces. II

This paper is an extension of [11]. Starting from the results of our first paper we prove by inclusion theorems that bounds for the correlation function of a stationary Gaussian process in the space of continuous functions with weight are strongly consistent and asymptotically normal. We construct the simplest functional confidence intervals in these spaces for the indeterminate correlation function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 322–329, March, 1991.     

Asymptotic properties of correlation bounds in functional spaces. II

This paper is an extension of [11]. Starting from the results of our first paper we prove by inclusion theorems that bounds for the correlation function of a stationary Gaussian process in the space of continuous functions with weight are strongly consistent and asymptotically normal. We construct the simplest functional confidence intervals in these spaces for the indeterminate correlation function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 322–329, March, 1991.     

Calculation of the asymptotics of the two-point correlation function for one-dimensional Bose gas

We consider a quantum field-theoretical model which describes spatially-nonhomogeneous, one-dimensional, repulsive Bose gas in an external harmonic potential. The two-point correlation function is calculated in the framework of functional integration. The corresponding functional integrals are estimated by means of stationary phase approximation. Asymptotic estimates are obtained in the limit as the temperature tends to zero while the volume occupied by the quasi-condensate increases. A power-law behavior is established for the correlation function in this limit. It is shown that the power-law behavior is governed by the critical exponent depending on spatial arguments. Bibliography 32 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 56–74.     

Central limit theorems for non-linear functionals of Gaussian fields

In the present paper it is shown that the central limit theorem holds for some non-linear functionals of stationary Gaussian fields if the correlation function of the underlying field tends fast enough to zero. The results are formulated in terms of the Hermite rank of the functional and of the rate of the correlation function. Then we show an example when the limit field is self-similar and Gaussian but not necessarily consisting of independent elements.     

Comparison theorems and asymptotic behavior of correlation estimators in spaces of continuous functions. II

This paper is the second part of [12]. Using the comparison theorems which were proved in the first part, the asymptotic normality of the estimator — in a model of a series of several samples — of the correlation function of a stationary Gaussian random process in spaces of continuous functions with weights is established. A method for constructing functional confidence intervals for an unknown correlation function in these spaces is described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 579–583, May, 1991.     

Functional Integration, Zeta Regularization, and Correlators of the Third Components of Spins in the Heisenberg XX0 Model

A new representation for the generating function of static correlators of the third components of spins in the XX0 Heisenberg model is suggested in the form of the Gaussian functional integral. The generating function, as well as the partition function of the model, are calculated by means of the zeta regularization procedure. Formulas for some correlation functions at nonzero temperature are obtained. Bibliography: 27 titles.     

Diagram theory for the periodic anderson model: Stationarity of the thermodynamic potential

We develop a diagram theory for the periodic Anderson model assuming that the Coulomb repulsion of localized f electrons is the main parameter of the theory. The f electrons are strongly correlated and the c conduction electrons are uncorrelated. We determine the f-electron correlation function and the c-electron mass operator. We formulate the Dyson equation for c electrons and a Dyson-type equation for f electrons and their propagators. We define the skeleton diagrams for the correlation function and the thermodynamic functional. We establish the stationarity of the renormalized thermodynamic potential under variation of the mass operator. The obtained results are applicable to both the normal and the superconducting system states.     

On asymptotic properties of the empirical correlation matrix of a homogeneous vector-valued Gaussian field

We investigate properties of the empirical correlation matrix of a centered stationary Gaussian vector field in various function spaces. We prove that, under the condition of integrability of the square of the spectral density of the field, the normalization effect takes place for a correlogram and integral functional of it.     

Asymptotic distributions of nonparametric regression estimators for longitudinal or functional data

The estimation of a regression function by kernel method for longitudinal or functional data is considered. In the context of longitudinal data analysis, a random function typically represents a subject that is often observed at a small number of time points, while in the studies of functional data the random realization is usually measured on a dense grid. However, essentially the same methods can be applied to both sampling plans, as well as in a number of settings lying between them. In this paper general results are derived for the asymptotic distributions of real-valued functions with arguments which are functionals formed by weighted averages of longitudinal or functional data. Asymptotic distributions for the estimators of the mean and covariance functions obtained from noisy observations with the presence of within-subject correlation are studied. These asymptotic normality results are comparable to those standard rates obtained from independent data, which is illustrated in a simulation study. Besides, this paper discusses the conditions associated with sampling plans, which are required for the validity of local properties of kernel-based estimators for longitudinal or functional data.     

A Functional Integral Representation for Many Boson Systems II: Correlation Functions

We derive functional integral representations for the partition function and correlation functions of many Boson systems for which the configuration space consists of finitely many points. Submitted: August 20, 2007. Accepted: July 1, 2008.     

Functional Data Model for Genetically Related Individuals With Application to Cow Growth

We propose a new version of functional data model for analyzing familial related individuals, where the within-subject correlation depends smoothly on a covariate such as age and the between-subject correlation follows family-wise genetic association. Our motivating example concerns measurements of weight as a function of age in sibling cows from independent families. Observations are sparsely sampled from trajectories of a phenotype contaminated with measurement error, where the phenotypic trajectory consists of a genetic component and an environmental component. By combining information across individuals, the genetic and environmental covariances are estimated via smoothing techniques. We study the genetic and environmental effects using principal component analysis, taking into account the genetic correlation to enhance the subject-level signal extraction. We show via the real data and simulations that incorporating the correlation structure improves predictions of individual phenotypic trajectories.     

Functional Additive Mixed Models

We propose an extensive framework for additive regression models for correlated functional responses, allowing for multiple partially nested or crossed functional random effects with flexible correlation structures for, for example, spatial, temporal, or longitudinal functional data. Additionally, our framework includes linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. It accommodates densely or sparsely observed functional responses and predictors which may be observed with additional error and includes both spline-based and functional principal component-based terms. Estimation and inference in this framework is based on standard additive mixed models, allowing us to take advantage of established methods and robust, flexible algorithms. We provide easy-to-use open source software in the pffr() function for the R package refund. Simulations show that the proposed method recovers relevant effects reliably, handles small sample sizes well, and also scales to larger datasets. Applications with spatially and longitudinally observed functional data demonstrate the flexibility in modeling and interpretability of results of our approach.     

Functional canonical analysis for square integrable stochastic processes

We study the extension of canonical correlation from pairs of random vectors to the case where a data sample consists of pairs of square integrable stochastic processes. Basic questions concerning the definition and existence of functional canonical correlation are addressed and sufficient criteria for the existence of functional canonical correlation are presented. Various properties of functional canonical analysis are discussed. We consider a canonical decomposition, in which the original processes are approximated by means of their canonical components.     

Direction selection in stochastic directional distance functions

Researchers rely on the distance function to model multiple product production using multiple inputs. A stochastic directional distance function (SDDF) allows for noise in potentially all input and output variables. Yet, when estimated, the direction selected will affect the functional estimates because deviations from the estimated function are minimized in the specified direction. Specifically, the parameters of the parametric SDDF are point identified when the direction is specified; we show that the parameters of the parametric SDDF are set identified when multiple directions are considered. Further, the set of identified parameters can be narrowed via data-driven approaches to restrict the directions considered. We demonstrate a similar narrowing of the identified parameter set for a shape constrained nonparametric method, where the shape constraints impose standard features of a cost function such as monotonicity and convexity.Our Monte Carlo simulation studies reveal significant improvements, as measured by out of sample radial mean squared error, in functional estimates when we use a directional distance function with an appropriately selected direction and the errors are uncorrelated across variables. We show that these benefits increase as the correlation in error terms across variables increase. This correlation is a type of endogeneity that is common in production settings. From our Monte Carlo simulations we conclude that selecting a direction that is approximately orthogonal to the estimated function in the central region of the data gives significantly better estimates relative to the directions commonly used in the literature. For practitioners, our results imply that selecting a direction vector that has non-zero components for all variables that may have measurement error provides a significant improvement in the estimator’s performance. We illustrate these results using cost and production data from samples of approximately 500 US hospitals per year operating in 2007, 2008, and 2009, respectively, and find that the shape constrained nonparametric methods provide a significant increase in flexibility over second order local approximation parametric methods.     

Sensitivity analysis of linear programming in the presence of correlation among right-hand side parameters or objective function coefficients

In the literature, sensitivity analysis of linear programming (LP) has been widely studied. However, only some very simple and special cases were considered when right-hand side (RHS) parameters or objective function coefficients (OFC) correlate to each other. In the presence of correlation when one parameter changes, other parameters vary, too. Here principal component analysis is used to convert the correlation of the LP homogenous parameters into functional relations. Then, using the derivatives of the functional relations, it is possible to perform classical sensitivity analysis for the LP with correlation among RHS parameters or OFC. The validity of the devised method is corroborated by open literature examples having correlation among homogenous parameters.     

Approximating multiple itô integrals with "band limited" processes

Let denote a "wide-band width" vector valued process. The paper is concerned with limits of , and for the ra-multiple integral case. For the most important case, a weak convergence result is obtained, and the "correction'" terms exhibited. The method is such that the conditions used can readily be weakened. An application to a likelihood functional and hypothesis testing problem is given. There, the weak convergence result (rather than mere convergence of finite dimensional distributions) is essential if the limit approximation is to make sense as an approximation to the likelihood functional. The correction terms depend only on the limit (as a-0) of the correlation function of the(renormalized)     

Functional Integration with an “Automorphic” Boundary Condition and Correlators of Third Components of Spins in the XX Heisenberg Model

For the generating function of static correlators of the third components of spins in the XX Heisenberg model, we derive a new representation given by a combination of Gaussian functional integrals over anticommuting variables. A peculiarity of the resulting functional integral is that a part of the integration variables depend on the imaginary time automorphically: these variables are multiplied by a certain complex number under the shift of the imaginary time by the period. The other variables satisfy the standard boundary conditions of the fermionic/bosonic type. Functional integration results are represented as determinants of matrix operators. We finally evaluate the generating function of correlators and the partition function of the model in the zeta-function regularization. The consistency of the suggested functional definition is confirmed by calculating several correlation functions of the third components of spins at a nonzero temperature.     

基于医学影像数据的大脑功能连接方法研究

本文采用不同的方法对比研究了精神病患者与 正常人大脑功能连接网络中的显著差异. 首先选用皮尔逊相关系数, 偏相关系数, 小波相关系数构造三种不同的大脑功能连接网络, 其次运用风险差检验出具有显著差异的功能连接, 最后进行比较和分析. 结果表明: 用不同方法得到的大脑功能连接网络中存在公共的异常连接. 在三种相关网络中本文均检验出病人的楔前叶与后扣带回这条连接较正常人而言功能减弱. 更值得注意的是, 一些公共异常连接的风险差值符号相反, 即病人大脑功能连接网络中该功能连接较正常人而言既增强又减弱, 表明在构造统计量寻找大脑功能异常连接时需作统计检验才能使结果更具说服力.     

The Generalized Moment Problem Associated with Correlation Measures

The classical power moment problem can be viewed as a theory of spectral representations of a positive functional on some classical commutative algebra with involution. We generalize this approach to the case in which the algebra is a special commutative algebra of functions on the space of multiple finite configurations. If the above-mentioned functional is generated by a measure on the space of finite ordinary configurations, then this measure is the correlation measure for a measure on the space of infinite configurations. The positiveness of the functional gives conditions for the measure to be a correlation measure.     

Spectral properties of Wick power series for a free field with an indefinite metric

The properties of infinite series in Wick powers of a free field whose two-point correlation function has singular infrared behavior and does not satisfy the positivity condition are investigated. If these series are defined on an appropriate functional domain, then the fields they converge to satisfy all conditions of the pseudo-Wightman formalism. For series convergent only on analytic test functions in the momentum representation, the spectral condition is formulated using the previously introduced notion of the carrier cone of an analytic functional. A suitable generalization of the Paley-Wiener-Schwartz theorem is used to prove that this condition is satisfied. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 57–73, October, 2000.